1. The General LISREL MODEL and Non-normality Ulf H. Olsson Professor of Statistics

2. Ulf H. Olsson Bivariate normal distribution

3. Ulf H. Olsson Positive vs. Negative Skewness Exhibit 1 These graphs illustrate the notion of skewness. Both PDFs have the same expectation and variance. The one on the left is positively skewed. The one on the right is negatively skewed.

4. Ulf H. Olsson Low vs. High Kurtosis Exhibit 1 These graphs illustrate the notion of kurtosis. The PDF on the right has higher kurtosis than the PDF on the left. It is more peaked at the center, and it has fatter tails.

5. Ulf H. Olsson Non-normality Skewness Kurtosis Ordinal Scale Interval Scale

6. Ulf H. Olsson Making Numbers S: sample covariance θ: parameter vector σ(θ): model implied covariance

7. Ulf H. Olsson Making Numbers

8. Ulf H. Olsson Making Numbers

9. Ulf H. Olsson Making Numbers

10. Ulf H. Olsson Making Numbers

11. Ulf H. Olsson Making Numbers Generally

12. Ulf H. Olsson Estimation 1) No AC provided ML or GLS 2) AC provided ML WLS (ADF) 3) Continuous or Ordinal

13. Ulf H. Olsson Ordinal Variables In practice, observed or measured variables are ofte ordinal However, ordinality is often ignored and numbers such as 1,2,3, etc. representing ordered categories, are treated as continuous variables. But, this is incorrect!

14. Ulf H. Olsson Estimation 1) No AC provided ML 2) AC provided and Polychoric Correlation WLS (ADF)

15. Ulf H. Olsson Examples Efficacy data From raw data to Model SPSS-file PSF-file